Estimates of Dirichlet eigenvalues of divergent elliptic operators in non-Lipschitz domains

Abstract: We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $\Omega \subset \mathbb C$. The suggested method is based on the quasiconformal composition operators on Sobolev spaces with applications to the weighted Poincar\'e-Sobolev inequalities.